Embedded newform 128.3.l.a.43.10

• Label: 128.3.l.a.43.10
• Level: $128$
• Weight: $3$
• Character: 128.43
• Analytic conductor: $3.488$
• Analytic rank: $0$
• Dimension: $496$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 128 = 2^{7} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	128.l (of order \(32\), degree \(16\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.48774738381\)
Analytic rank:	\(0\)
Dimension:	\(496\)
Relative dimension:	\(31\) over \(\Q(\zeta_{32})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{32}]$

Embedding invariants

Embedding label			43.10
Character	\(\chi\)	\(=\)	128.43
Dual form			128.3.l.a.3.10

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.31385 - 1.50791i) q^{2} +(-2.45029 - 2.01090i) q^{3} +(-0.547604 + 3.96234i) q^{4} +(9.14318 - 2.77355i) q^{5} +(0.187046 + 6.33684i) q^{6} +(0.716485 - 3.60202i) q^{7} +(6.69433 - 4.38017i) q^{8} +(0.204383 + 1.02750i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 496 q - 16 q^{2} - 16 q^{3} - 16 q^{4} - 16 q^{5} - 16 q^{6} - 16 q^{7} - 16 q^{8} - 16 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/128\mathbb{Z}\right)^\times\).

\(n\)	\(5\)	\(127\)
\(\chi(n)\)	\(e\left(\frac{29}{32}\right)\)	\(-1\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

(See \(a_n\) instead)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.31385	−	1.50791i	−0.656924	−	0.753957i
							\(3\)	−2.45029	−	2.01090i	−0.816763	−	0.670300i	0.130255	−	0.991481i	\(-0.458420\pi\)
−0.947018	+	0.321180i	\(0.895920\pi\)
\(5\)	9.14318	−	2.77355i	1.82864	−	0.554711i	0.828649	−	0.559769i	\(-0.189110\pi\)
							0.999987	+	0.00505841i	\(0.00161015\pi\)
\(7\)	0.716485	−	3.60202i	0.102355	−	0.514574i	−0.895260	−	0.445545i	\(-0.853010\pi\)
							0.997615	−	0.0690287i	\(-0.0219900\pi\)
\(11\)	0.982186	−	9.97230i	0.0892897	−	0.906573i	−0.841315	−	0.540546i	\(-0.818218\pi\)
							0.930604	−	0.366027i	\(-0.119282\pi\)
\(13\)	−6.88056	+	22.6822i	−0.529274	+	1.74478i	0.128644	+	0.991691i	\(0.458938\pi\)
							−0.657917	+	0.753090i	\(0.728562\pi\)
\(17\)	−5.22340	−	12.6104i	−0.307259	−	0.741788i	−0.999792	−	0.0204023i	\(-0.993505\pi\)
							0.692533	−	0.721386i	\(-0.256495\pi\)
\(19\)	−11.3592	−	21.2516i	−0.597854	−	1.11851i	−0.980852	−	0.194757i	\(-0.937608\pi\)
							0.382998	−	0.923749i	\(-0.374892\pi\)
\(23\)	6.68053	+	4.46378i	0.290458	+	0.194078i	0.692259	−	0.721649i	\(-0.256616\pi\)
							−0.401801	+	0.915727i	\(0.631616\pi\)
\(29\)	−0.792725	−	8.04867i	−0.0273353	−	0.277540i	−0.999130	−	0.0417106i	\(-0.986719\pi\)
							0.971794	−	0.235830i	\(-0.0757808\pi\)
\(31\)	3.89963	−	3.89963i	0.125795	−	0.125795i	−0.641407	−	0.767201i	\(-0.721649\pi\)
							0.767201	+	0.641407i	\(0.221649\pi\)
\(37\)	17.2810	−	32.3304i	0.467053	−	0.873796i	−0.532571	−	0.846385i	\(-0.678774\pi\)
							0.999624	−	0.0274101i	\(-0.00872600\pi\)
\(41\)	−11.4822	−	7.67219i	−0.280055	−	0.187127i	0.407608	−	0.913157i	\(-0.366363\pi\)
							−0.687662	+	0.726031i	\(0.741363\pi\)
\(43\)	−16.0981	+	13.2114i	−0.374374	+	0.307241i	−0.802700	−	0.596384i	\(-0.796604\pi\)
							0.428325	+	0.903625i	\(0.359104\pi\)
\(47\)	26.8031	+	64.7085i	0.570280	+	1.37678i	0.901317	+	0.433159i	\(0.142601\pi\)
							−0.331038	+	0.943618i	\(0.607399\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	128.3.l.a.43.10	yes	496
128.3	odd	32	inner	128.3.l.a.3.10	✓	496

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
128.3.l.a.3.10	✓	496	128.3	odd	32	inner
128.3.l.a.43.10	yes	496	1.1	even	1	trivial